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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 6 — Mar. 14, 2011
  • pp: 5658–5669
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Quasi-two-dimensional optomechanical crystals with a complete phononic bandgap

Thiago P. Mayer Alegre, Amir Safavi-Naeini, Martin Winger, and Oskar Painter  »View Author Affiliations


Optics Express, Vol. 19, Issue 6, pp. 5658-5669 (2011)
http://dx.doi.org/10.1364/OE.19.005658


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Abstract

A fully planar two-dimensional optomechanical crystal formed in a silicon microchip is used to create a structure devoid of phonons in the GHz frequency range. A nanoscale photonic crystal cavity is placed inside the phononic bandgap crystal in order to probe the properties of the localized acoustic modes. By studying the trends in mechanical damping, mode density, and optomechanical coupling strength of the acoustic resonances over an array of structures with varying geometric properties, clear evidence of a complete phononic bandgap is shown.

© 2011 Optical Society of America

1. Introduction

Photonic crystals are periodic dielectric structures in which optical waves encounter strong dispersion, and in some cases are completely forbidden from propagating within a frequency bandgap. Similarly, periodic elastic structures, known as phononic crystals, can be used to control acoustic waves [1

1. Z. Liu, X. Zhang, Y. Mao, Y. Y. Zhu, Z. Yang, C. T. Chan, and P. Sheng, “Locally Resonant Sonic Materials,” Science 289(5485), 1734–1736 (2000). [CrossRef] [PubMed]

7

7. Y. Wen, J. Sun, C. Dais, D. Grtzmacher, T. Wu, J. Shi, and C. Sun, “Three-dimensional phononic nanocrystal composed of ordered quantum dots,” Appl. Phys. Lett. 96(12), 123113 (2010). [CrossRef]

] for applications as diverse as the filtering and focusing of sound [8

8. S. Yang, J. H. Page, Z. Liu, M. L. Cowan, C. T. Chan, and P. Sheng, “Focusing of Sound in a 3D Phononic Crystal,” Phys. Rev. Lett. 93(2), 024301 (2004). [CrossRef] [PubMed]

] to the earthquake proofing of buildings [9

9. J. Gaofeng and S. Zhifei, “A new seismic isolation system and its feasibility study,” Earthq. Eng. Eng. Vib. 9(1), 75–82 (2010). [CrossRef]

]. Recently, microchip structures capable of manipulating both photons and phonons, dubbed optomechanical crystals (OMCs), have been created to enhance the interaction of light and mechanics [10

10. M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462(7269), 78–82 (2009). [CrossRef] [PubMed]

15

15. Y. Pennec, B. D. Rouhani, E. H. El Boudouti, C. Li, Y. El Hassouani, J. O. Vasseur, N. Papanikolaou, S. Bench-abane, V. Laude, and A. Martinez, “Simultaneous existence of phononic and photonic band gaps in periodic crystal slabs,” Opt. Express 18(13), 14301–14310 (2010). [CrossRef] [PubMed]

]. Here we use a planar two-dimensional (2D) OMC to create a structure devoid of phonons in the GHz frequency range, and use light to probe its mechanical properties. In addition to being an excellent platform for the study of radiation pressure and nanomechanics, the 2D-OMC of this work represents an important step towards more complex devices capable of performing novel classical and quantum optical signal processing [16

16. D. Chang, A. H. Safavi-Naeini, M. Hafezi, and O. Painter, “Slowing and stopping light using an optomechanical crystal array,” arXiv:1006.3829 (2010).

, 17

17. A. H. Safavi-Naeini and O. Painter, “Proposal for an Optomechanical Traveling Wave Phonon-Photon Translator,” New J. Phys. 13, 013017 (2011). [CrossRef]

].

2. Sample structure

The phononic crystal used in this work is the recently proposed [14

14. A. H. Safavi-Naeini and O. Painter, “Design of optomechanical cavities and waveguides on a simultaneous bandgap phononic-photonic crystal slab,” Opt. Express 18(14), 14926–14943 (2010). [CrossRef] [PubMed]

] “cross” structure shown in Fig. 1(a). Geometrically, the structure consists of an array of squares connected to each other by thin bridges, or equivalently, a square lattice of cross-shaped holes. The phononic bandgap in this structure arises from the frequency separation between higher frequency tight-binding bands, which have similar frequencies to the resonances of the individual squares, and lower frequency effective-medium bands with frequencies strongly dependent on the width of the connecting bridges, b = a – h [14

14. A. H. Safavi-Naeini and O. Painter, “Design of optomechanical cavities and waveguides on a simultaneous bandgap phononic-photonic crystal slab,” Opt. Express 18(14), 14926–14943 (2010). [CrossRef] [PubMed]

]. A typical band diagram for a nominal structure (a = 1.265 μm, h = 1.220 μm, w = 340 nm) is shown in Fig. 1(c). Blue (red) lines represent bands with even (odd) vector symmetry for reflections about the xy plane. The lowest frequency bandgap for the even modes of the simulated cross structure extends from 0.91 GHz to 3.6 GHz. Within this bandgap, there are regions of full phononic bandgap (shaded blue) where no mechanical modes of any symmetry exist, and regions of partial symmetry-dependent bandgap (shaded red) where out-of-plane flexural modes with odd symmetry about the xy plane are allowed. As the bridge width is increased, the lower frequency effective-medium bands become stiffer, causing an increase in their frequency, while the higher frequency tight-binding band frequencies remain essentially constant. A gap-map show in in Fig. 1(d), showing how the bandgaps in the structure change as a function of phononic crystal bridge width, illustrates this general feature.

Fig. 1 (a) Real space crystal lattice of the cross crystal with lattice constant a, cross length h, cross width w, and membrane thickness d. The bridge width is defined as b = a – h. (b) Reciprocal lattice of the first Brillouin zone for the cross crystal. (c) Phononic band diagram for the nominal cross structure with a = 1.265 μm, h = 1.220 μm, w = 340 nm. Dark blue lines represent the bands with even vector symmetry for reflections about the x – y plane, while the red lines are the flexural modes which have odd vector mirror symmetry about the x – y plane. (d) Tuning of the bandgap with bridge width, b. Light grey, dark grey, and white areas indicate regions of a symmetry-dependent (i.e., for modes of only one symmetry) bandgap, no bandgap, and full bandgap for all acoustic modes, respectively.

As shown in Figs. 2(a)–(c), the cross crystal is used as a phononic cage (cavity) for an embedded optical nanocavity [28

28. B. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater 4(3), 207–210 (2005). [CrossRef]

] (highlighted in a green false color) consisting of a quasi-2D photonic crystal waveguide with a centralized “defect” region for localizing photons. This embedding of an optical cavity within an acoustic cavity enables, through the strong radiation-pressure-coupling of optical and acoustic waves, the probing of the properties of the bandgap-localized phonons via a light field sent through the optical nanocavity. The theoretical electric field mode profile and the measured high-Q nature of the optical resonance of the photonic crystal cavity are shown in Figs. 2(d) and (e), respectively. Such a phonon-photon heterostructure design allows for completely independent tuning of the mechanical and optical properties of our system, and in what follows, we use this feature to probe arrays of structures with different geometric parameters. In particular, by varying the bridge width b of the outer phononic bandgap crystal, the lower bandgap edge can be swept in frequency and the resulting change in the lifetime, density of states, and localization of the trapped acoustic waves interacting with the central optical cavity can be monitored. Two different phonon cavity designs, S1 and S2, were fabricated in this study. First, in Sec. 5.1, we focus on the lower acoustic frequency S1 structure, for which an array of devices with bridge width varying from b = 53 nm to 173 nm (in 6 nm increments) was created. Similar results are also shown for the high acoustic frequency S2 structure in Sec. 5.2.

Fig. 2 (a) Scanning electron micrograph (SEM) of one of the fabricated 2D-OMC structures. The photonic nanocavity region is shown in false green color. In (b), Zoom-in SEM image of the cross crystal phononic bandgap structure. (c) Zoom-in SEM image of the optical nanocavity within embedded in the phononic bandgap crystal. Darker (lighter) false colors represents larger (smaller) lattice constant in the optical cavity defect region. (d) FEM simulation of Ey electrical field for the optical cavity. (e) Typical measured transmission spectra for the optical nanocavity, showing a bare optical Q-factor of Qi = 1.5 × 106.

3. Fabrication

The phononic-photonic cavities were fabricated using a Silicon-On-Insulator wafer from SOITEC (ρ = 4–20 Ω·cm, device layer thickness t = 220 nm, buried-oxide layer thickness 2 μm). The cavity geometry is defined by electron beam lithography followed by inductively-coupled-plasma reactive ion etching (ICP-RIE) to transfer the pattern through the 220 nm silicon device layer. The cavities were then undercut using HF:H2O solution to remove the buried oxide layer, and cleaned using a piranha/HF cycle [29

29. M. Borselli, T. J. Johnson, and O. Painter, “Measuring the role of surface chemistry in silicon microphotonics,” Appl. Phys. Lett. 88(13), 131,114–3 (2006). [CrossRef]

].

4. Experimental setup

The experimental setup used to measure the phononic-photonic crystal cavity properties is shown in Fig. 3(a). A fiber-coupled tunable near-infrared laser, (New Focus Velocity, model TLB-6328) spanning approximately 60 nm, centered around 1540 nm, has its intensity and polarization controlled respectively by a variable optical attenuator (VOA) and a fiber polarization controller (FPC). The laser light is coupled to a tapered, dimpled optical fiber (Taper) which has its position controlled with nanometer-scale precision. The transmission from the fiber is passed through another VOA before being detected.

Fig. 3 (a) Experimental setup for measuring the PSD. (b) Optical micrograph of the tapered fiber coupled to the device while performing experiments. (c) Optical spectra for two different positions of the taper relative to the device are shown.

To measure the optical properties, a photodetector (PD, New Focus Nanosecond Photodetector, model 1623) is used. The detected optical transmission signal is recorded while sweeping the laser frequency. By controlling the distance between the fiber taper and the sample, the external coupling rate (κe) is changed. Figure 3(c) shows the change in the coupling rate for two different positions of the fiber taper. In the limit where the external coupling rate is zero we can measure the intrinsic coupling rate (κi). The total optical loss is then κ = κe + κi. A typical fiber taper transmission spectrum is shown in Fig. 2(e), with a measured intrinsic optical quality factor of Qi = 1.5 × 106. Usually, after touching, the external coupling rate was on the order of tens of MHz (κe/2π ≈ 70 MHz) which corresponds to a transmission dip of ≈ 70%.

To measure the mechanical properties, the transmitted signal is sent through an erbium doped fiber amplifier (EDFA) and sent to a high-speed photoreceiver (PR, New Focus model, 1554-B) with a maximum transimpedance gain of 1,000 V/A and a bandwidth (3 dB rolloff point) of 12 GHz. The RF voltage from the photoreceiver is connected to the 50 Ω input impedance of the oscilloscope. The oscilloscope can perform a Fast Fourier Transform (FFT) to yield the RF power spectral density (RF PSD). The RF PSD is calibrated using a frequency generator that outputs a variable frequency sinusoid with known power.

Our devices are in the sideband resolved limit, i.e. the total optical loss rate is smaller than the mechanical frequency, κ < ΩM. Therefore the largest transduced signal is achieved when the laser frequency is detuned from the optical cavity resonance by the mechanical frequency [10

10. M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462(7269), 78–82 (2009). [CrossRef] [PubMed]

]. The probe laser is locked to approximately 1 GHz (2.5 GHz) on the blue side of the cavity resonance for the S1 (S2) structures. By measuring the transmission contrast during the acquisition of the RF PSD and comparing with the transmission curve of each device (as shown in Fig. 3(c)) we determine the laser detuning and the dropped power into the cavity. To lock the probe laser frequency a given frequency away from the resonance, a 90/10 beam splitter (BS) is added to the optical path, and the signal from the 10% arm is fed to a PD connected to a locking circuit which compares the voltage level from the transmission signal to a predetermined value to generate an error signal. The error signal is then fed to a piezoelectric actuator on the feedback grating of the laser in order to stabilize the laser frequency.

5. Results and discussion

5.1. Phononic band gap: sample S1

Experimentally we observe the thermally excited acoustic modes of the photonic-phononic crystal through the induced phase-modulation of the optical cavity field. The mixing of the phase-modulated light from the optical cavity with the transmitted light produces RF/microwave tones upon optical detection with a high speed photodetector. The measured RF-spectra from three different S1 structures with small (b = 57 nm), medium (b = 106 nm), and large (b = 160 nm) bridge widths are shown in Fig. 4(a). Each narrow tone in the RF-spectra corresponds to a different acoustic resonance interacting with the central optical nanocavity. Through careful calibration of the optical power and electronic detection, one can extract both the mechanical Q-factor (from the linewidth) and the level of optomechanical coupling (from the magnitude of the transduced thermal motion) of each acoustic resonance. Here we parametrize the strength of the optomechanical coupling by the rate g, which corresponds physically to the shift in the optical cavity resonance frequency due to the zero-point motion of the acoustic resonance [14

14. A. H. Safavi-Naeini and O. Painter, “Design of optomechanical cavities and waveguides on a simultaneous bandgap phononic-photonic crystal slab,” Opt. Express 18(14), 14926–14943 (2010). [CrossRef] [PubMed]

]. A measurement of the temperature dependence of the acoustic mode spectrum is also performed, and is shown in Fig. 4(b) for one of the acoustic resonances over a temperature range from 300 K to 10 K.

Fig. 4 (a) Optically transduced RF power spectral density of the thermal Brownian motion of S1 structures with b = 57 nm (top), b = 106 nm (middle), and b = 160 nm (bottom). The inferred region below the phononic bandgap is shaded grey (see main text). (b) Temperature dependence of the mechanical quality factor for the 1.4 GHz acoustic mode of the S1 structure with b = 57 nm.

By measuring the entire set of S1 devices in this way, a map may be produced of the localized acoustic modes’ properties versus bridge width. In Figs. 5(a) and 5(b) we plot the numerically simulated and experimentally measured mode map for the S1 structure. Each marker in these plots corresponds to a different acoustic resonance, with the position of the marker indicating the mode frequency and the size of the marker indicating the mechanical Q-factor of the mode (for the numerical simulations all mechanical Q-factors above 107 are shown with the same marker size). Numerical simulations of the optical, mechanical, and optomechanical properties of the structure are performed using the COMSOL [30

30. COMSOL Multphysics 3.5 (2009).

] finite-element-method (FEM) software package, with an absorbing boundary condition applied at the exterior of the phononic cage [12

12. M. Eichenfield, J. Chan, A. H. Safavi-Naeini, K. J. Vahala, and O. Painter, “Modeling dispersive coupling and losses of localized optical and mechanical modes in optomechanical crystals,” Opt. Express 17(22), 20078–20098 (2009). [CrossRef] [PubMed]

]. The various bandgap regions are indicated in Fig. 5(a) with the same color coding as in Fig. 1(c). Due to the weak radiation pressure coupling to the optical nanocavity of the flexure acoustic modes of odd symmetry about the xy plane of the slab (red mode bands in Fig. 1(c)), we only show in the simulated mode plot of Fig. 5(a) the even symmetry, in-plane acoustic resonances.

Fig. 5 (a) Plot of the 3D-FEM simulated in-plane localized acoustic modes of the S1 structure as a function of bridge width b. Each marker corresponds to a single acoustic mode, with the marker size proportional to the logarithm of the calculated acoustic radiation Q-factor. The light blue shaded markers correspond to acoustic bands which are optically dark. The shading corresponds to the same color coding of the phononic bandgaps as that used in Fig. 1(d). (b) Measured mode plot of the optically-transduced localized acoustic modes for an array of S1 structures with varying bridge width. The marker size of each resonance is related to the logarithm of the measured mechanical Q-factor. The inferred spectral region below the phononic bandgap is shaded grey. (c) and (d) FEM simulations of the displacement field amplitude (|Q(r)|) for the acoustic mode in the orange colored band around 1.35 GHz in (a). In (c) the mode is within the phononic band gap resulting in a radiation-limited QM(rad)109. In (d) the mode is on the edge of the bandgap and has a reduced QM(rad)<103. (e) Simulated (□) and measured (○) optomechanical coupling rate g for the orange (red) highlighted acoustic band in (a) ((b)). (f) Simulated (□) and measured (○) optomechanical coupling rate g for the series of acoustic modes of the S1 structure with b ∼ 100 nm (vertical dashed curves in (a) and (b))

The striking similarity of the simulated and measured mode plots is evidence that the optical nanocavity is able to sensitively probe the in-plane localized acoustic modes of the phononic bandgap structure (the acoustic band with light blue marker in Fig. 5(a) is the one localized in-plane mode which does not show up in the measured plot of Fig. 5(b); numerical simulations show this mode to be a surface mode at the inner edge of the cross crystal, which does not couple to the central optical cavity). Within the bandgap, modes are tightly localized (see Fig. 5(c)) and do not radiate acoustic energy, whereas below the bandgap the acoustic modes spread into the exterior cross crystal (see Fig. 5(d)), leaking energy into the surrounding substrate region. The boundary where the mechanical Q-factor drops off is clearly identifiable in the experimentally measured mode plot of Fig. 5(b) (the spectral region below the apparent full phononic bandgap is shaded grey as a guide to the eye), and matches up well with the theoretical lower frequency band-edge of the full phononic bandgap of the cross crystal.

Two other distinguishing features between modes inside and outside a bandgap are the spectral mode density and the strength of the optomechanical coupling. Below the phononic bandgap, acoustic modes can fill the entire volume of the cross crystal (out to the boundary of the undercut structure where it is finally clamped), resulting in an increase of the mode density (proportional to volume) and a decrease in the optomechanical coupling (proportional to the inverse-square-root of mode volume [14

14. A. H. Safavi-Naeini and O. Painter, “Design of optomechanical cavities and waveguides on a simultaneous bandgap phononic-photonic crystal slab,” Opt. Express 18(14), 14926–14943 (2010). [CrossRef] [PubMed]

]). In Fig. 5(e) we have plotted the theoretically computed and experimentally measured values of the optomechancial coupling (g) for an acoustic resonance lying near the middle of the full phononic bandgap (this resonance is highlighted in orange in the theoretical plot of Fig. 5(a) and red in the measured plot of Fig. 5(b)). The measured trend of optomechanical coupling nicely matches that of the theoretical one, and highlights the sharp drop off in optomechanical coupling as the mode crosses the bandgap. Similarly, in Fig. 5(f) we plot the optomechanical coupling for the acoustic resonances of a single device with bridge width b ∼ 100 nm (corresonding to a vertical slice in Figs. 5(a) and 5(b) as indicated by a dashed vertical line). Again we see good correspondence between theory and experiment, with the drop off in g and the large increase in spectral mode density clearly evident in both plots below the bandgap (note that the frequency position of the bandgap edge is not the same in the theoretical and experimental plots of Figs. 5(e) and 5(f) due to the slight differences in bridge width).

Finally we note the high sensitivity of the most strongly transduced modes, by analyzing the spectra shown in Fig. 4. The room temperature sensitivity of the mechanical mode was calculated to be 5.0×1017m/Hz. This sensitivity increases to 2.5×1017m/Hz at low temperatures (8.7 K), due to a four-fold increase in mechanical Q.

5.2. Phononic band gap: sample S2

In this section we present measurements performed on a second design (S2) with phononic bandgap in a different frequency range. Following the nomenclature in Fig. 1, the nominal dimensions for the S2 devices are: a = 925 nm, h = 850 nm, and w = 210 nm, which allows for an acoustic bandgap around 1.6 to 2.5 GHz. The optical nanocavity in the S2 devices are the same as that used in S1 devices.

Fig. 6 (a) RF-power spectral density for three devices with different bridge widths, b. The gray shaded area refers to regions outside of the full bandgap while the orange shaded area are for frequencies within the bandgap. (b) results of full 3D-FEM simulations of phononic localized modes as a function of bridge width b. For each bridge-width a full 3D-FEM simulation with an absorbing perfect-matched-layer (PML) is performed. Each square corresponds to a single mode, with its size is proportional to the logarithm of the mechanical Q-factor. The gray shaded squares represent modes which are optically dark (g’s small enough to be lower than our detection noise floor). (c) Measurement (○) results of the localized mechanical modes for the a set of fabricated S2 samples. Each bridge size b was measured by careful analysis of scanning electron micrographs. The marker sizes are proportional to measured mechanical Q-factor.

6. Mechanical damping

Having localized acoustic modes via a full 3D phononic bandgap, and at least in principle removed radiation losses, it is interesting to consider the limits to mechanical damping in these structures. As shown in Fig. 4(b) for the in-plane acoustic resonance at 1.40 GHz lying well within the phononic bandgap of the cross crystal, the mechanical Q-factor increases from a value just above 3000 at room temperature to a value of 1.3 × 104 at a temperature of 10 K. For micro- and nano-mechanical structures the many loss mechanism include clamping losses, dissipation due to phonon-phonon interactions, and surface effects. In Fig. 7(a), we compare the temperature dependence of mechanical Q to Akheiser [31

31. A. Akhieser, J. Phys. (Moscow) 1, 277 (1939).

] damping, Landau-Rumer [32

32. L. Landau and G. Rumer, “Absorption of sound in solids,” Phys. Z. Sowjetunion 11, 18 (1937).

] damping, a numerical model of thermoelastic damping in the structure [33

33. C. Zener, “Internal Friction in Solids II. General Theory of Thermoelastic Internal Friction,” Phys. Rev. 53(1), 90 (1938). [CrossRef]

], and measurements of acoustic wave attenuation in bulk Si [34

34. S. D. Lambade, G. G. Sahasrabudhe, and S. Rajagopalan, “Temperature dependence of acoustic attenuation in silicon,” Phys. Rev. B 51(22), 15,861 (1995). [CrossRef]

] with our results (blue circles). For the acoustic attenuation, α(Ω; T), the relevant bulk measurement results are taken from Ref. [34

34. S. D. Lambade, G. G. Sahasrabudhe, and S. Rajagopalan, “Temperature dependence of acoustic attenuation in silicon,” Phys. Rev. B 51(22), 15,861 (1995). [CrossRef]

] and we plot (black diamonds) the calculated QM = 2πΩM/(2α), where ΩM/2π = 1.4 GHz is the mechanical frequency. The measured values from Ref. [34

34. S. D. Lambade, G. G. Sahasrabudhe, and S. Rajagopalan, “Temperature dependence of acoustic attenuation in silicon,” Phys. Rev. B 51(22), 15,861 (1995). [CrossRef]

] indicate that our devices are not limited by bulk losses.

Fig. 7 (a) Comparison between the different sources of mechanical loss with the measured QM values versus temperature. The circles represent the measured values from the S1 samples; diamonds are computed QM from acoustic attenuation data from Ref. [34]; squares represent the simulated values for TED as explained on the text. The purple and green lines are the calculated QM due to Akheiser and Landau-Rumer phonon-phonon dissipation mechanism respectively. The insets show the measured RF PSD at 10 K and 300 K for extracting QM. (b) Thermo-mechanical 2D-FEM simulations for the mechanical mode at 1.4 GHz shown in Fig. 3 of the main text. The thermal profile is plotted at various times during the mechanical cycle.

In order to provide an upper bound to the quality factor in our structures, we compare our measurements with the theoretical temperature-dependent acoustic attenuation provided by the Akheiser [31

31. A. Akhieser, J. Phys. (Moscow) 1, 277 (1939).

] (green dashed line), Landau-Rumer [32

32. L. Landau and G. Rumer, “Absorption of sound in solids,” Phys. Z. Sowjetunion 11, 18 (1937).

] (purple dashed line), and Thermoelastic Damping [33

33. C. Zener, “Internal Friction in Solids II. General Theory of Thermoelastic Internal Friction,” Phys. Rev. 53(1), 90 (1938). [CrossRef]

, 35

35. C. Zener, “Internal Friction in Solids. I. Theory of Internal Friction in Reeds,” Phys. Rev. 52(3), 230 (1937). [CrossRef]

, 36

36. R. Lifshitz and M. L. Roukes, “Thermoelastic damping in micro- and nanomechanical systems,” Phys. Rev. B 61(8), 5600 (2000). [CrossRef]

] (TED - red square points) models. The Landau-Rumer model provides a microscopic theory for sound absorption and is valid on the limit of ΩMτ ≫ 1, where τ is the mean time between collisions of thermal-phonon in the solid. Akheiser’s model treats the dissipation of heat generated by strain trough the Boltzmann transport equation [37

37. T. O. Woodruff and H. Ehrenreich, “Absorption of Sound in Insulators,” Phys. Rev. 123(5), 1553 (1961). [CrossRef]

] and is valid for ΩMτ ≪ 1. Finally TED considers the heat diffusion in homogeneous materials which can be calculated for any geometry.

The equations for acoustic attenuation for Landau-Rumer (αLR) and Akheiser’s (αAK) models used are [37

37. T. O. Woodruff and H. Ehrenreich, “Absorption of Sound in Insulators,” Phys. Rev. 123(5), 1553 (1961). [CrossRef]

]:
αLR(Ω;T)=πγ2ΩCvT4ρcs2,andαAK(Ω;T)=γ2Ω2CvTτ3ρcs2,
(1)
where γ is the average Grüneisen coefficient extract from Ref. [38

38. J. Philip and M. A. Breazeale, “Third-order elastic constants and Grüneisen parameters of silicon and germanium between 3 and 300 K,” J. Appl. Phys. 54(2), 752 (1983). [CrossRef]

], Cv is the volumetric heat capacity with values from Ref. [39

39. S. K. Estreicher, M. Sanati, D. West, and F. Ruymgaart, “Thermodynamics of impurities in semiconductors,” Phys. Rev. B 70(12), 125209 (2004). [CrossRef]

], and ρ = 2330 kg/m3 and cs = 9.15 × 103 m/s are the density and average speed of sound of Si respectively. Figure 7(a) shows that for temperatures above T = 100 K the Akheiser and Landau-Rumer losses are dominant. Note that ΩMτ = 1 for T ≅ 200 K and only Landau-Rumer is valid below this point. In this region TED effects due to the sample geometry become important.

Our approach for calculating TED follows that of Ref. [40

40. A. Duwel, R. Candler, T. Kenny, and M. Varghese, “Engineering MEMS Resonators With Low Thermoelastic Damping,” J. Microelectromech. Syst. 15(6), 1437–1445 (2006). [CrossRef]

], where the TED-limited QM,TED is extracted from 2D-FEM simulations [30

30. COMSOL Multphysics 3.5 (2009).

] for the thermo-mechanical equations considering a finite thickness. Figure 7(b) shows the change in temperature, ΔT (r) = T – To, for the deformed structure, versus the phase of the mechanical oscillations, at the expected brownian motion amplitude for To = 300 K. The temperature difference was calculated based on the maximum thermal displacement amplitude xmax=2kTo/meffΩM2. These plots show that during a mechanical cycle, even when there is no deformation, i.e. ΩMt = π/2 and 3π/2, the temperature gradient is non-zero. This indicates that the temperature does not follow adiabatically the strain/stress profile, causing a time-delayed force to be imparted on the resonator, and leading to dissipation.

The measured temperature dependence and the overall magnitude of the measured mechanical Q-factor are seen to be much smaller than any of these comparisons, suggesting that surface effects and/or fabrication-induced damage may be playing an important role in the mechanical damping of the nanostructured devices studied here.

7. Conclusion

Beyond the confinement and localization of acoustic modes in three dimensions, the connected geometry of the 2D-OMC structures presented in this work offers a platform for more complex phonon-photon circuitry. As has been described in recent theoretical analyses [14

14. A. H. Safavi-Naeini and O. Painter, “Design of optomechanical cavities and waveguides on a simultaneous bandgap phononic-photonic crystal slab,” Opt. Express 18(14), 14926–14943 (2010). [CrossRef] [PubMed]

, 16

16. D. Chang, A. H. Safavi-Naeini, M. Hafezi, and O. Painter, “Slowing and stopping light using an optomechanical crystal array,” arXiv:1006.3829 (2010).

, 17

17. A. H. Safavi-Naeini and O. Painter, “Proposal for an Optomechanical Traveling Wave Phonon-Photon Translator,” New J. Phys. 13, 013017 (2011). [CrossRef]

], such circuitry could be used to create optomechanical systems with greatly enhanced optomechanical coupling, and to realize devices such as traveling wave phonon-photon translators and slow light waveguides capable of advanced classical and quantum optical signal processing. The functionality of these devices are based upon the slow propagation velocity and long relative lifetime of phonons in comparison to photons, which allows for the storage, buffering, and narrowband filtering of optical signals. In addition, the coupling of optomechanical circuits to a wide variety other physical systems, such as superconducting electronic circuits and atomic vapors, may also enable the interfacing and networking of different quantum systems.

Acknowledgments

This work was supported by the DARPA/MTO ORCHID program through a grant from AFOSR, and the Kavli Nanoscience Institute at Caltech. ASN gratefully acknowledges support from NSERC.

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W. Cheng, J. Wang, U. Jonas, G. Fytas, and N. Stefanou, “Observation and tuning of hypersonic bandgaps in colloidal crystals,” Nat. Mater. 5(10), 830–836 (2006). [CrossRef] [PubMed]

4.

A. V. Akimov, Y. Tanaka, A. B. Pevtsov, S. F. Kaplan, V. G. Golubev, S. Tamura, D. R. Yakovlev, and M. Bayer, “Hypersonic Modulation of Light in Three-Dimensional Photonic and Phononic Band-Gap Materials,” Phys. Rev. Lett. 101(3), 033902 (2008). [CrossRef] [PubMed]

5.

T. Gorishnyy, C. K. Ullal, M. Maldovan, G. Fytas, and E. L. Thomas, “Hypersonic Phononic Crystals,” Phys. Rev. Lett. 94(11), 115501 (2005). [CrossRef] [PubMed]

6.

S. Mohammadi, A. A. Eftekhar, W. D. Hunt, and A. Adibi, “High-Q micromechanical resonators in a two-dimensional phononic crystal slab,” Appl. Phys. Lett. 94(5), 051906 (2009). [CrossRef]

7.

Y. Wen, J. Sun, C. Dais, D. Grtzmacher, T. Wu, J. Shi, and C. Sun, “Three-dimensional phononic nanocrystal composed of ordered quantum dots,” Appl. Phys. Lett. 96(12), 123113 (2010). [CrossRef]

8.

S. Yang, J. H. Page, Z. Liu, M. L. Cowan, C. T. Chan, and P. Sheng, “Focusing of Sound in a 3D Phononic Crystal,” Phys. Rev. Lett. 93(2), 024301 (2004). [CrossRef] [PubMed]

9.

J. Gaofeng and S. Zhifei, “A new seismic isolation system and its feasibility study,” Earthq. Eng. Eng. Vib. 9(1), 75–82 (2010). [CrossRef]

10.

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462(7269), 78–82 (2009). [CrossRef] [PubMed]

11.

M. Maldovan and E. L. Thomas, “Simultaneous localization of photons and phonons in two-dimensional periodic structures,” Appl. Phys. Lett. 88(25), 251907 (2006). [CrossRef]

12.

M. Eichenfield, J. Chan, A. H. Safavi-Naeini, K. J. Vahala, and O. Painter, “Modeling dispersive coupling and losses of localized optical and mechanical modes in optomechanical crystals,” Opt. Express 17(22), 20078–20098 (2009). [CrossRef] [PubMed]

13.

S. Mohammadi, A. A. Eftekhar, A. Khelif, and A. Adibi, “Simultaneous two-dimensional phononic and photonic band gaps in opto-mechanical crystal slabs,” Opt. Express 18(9), 9164–9172 (2010). [CrossRef] [PubMed]

14.

A. H. Safavi-Naeini and O. Painter, “Design of optomechanical cavities and waveguides on a simultaneous bandgap phononic-photonic crystal slab,” Opt. Express 18(14), 14926–14943 (2010). [CrossRef] [PubMed]

15.

Y. Pennec, B. D. Rouhani, E. H. El Boudouti, C. Li, Y. El Hassouani, J. O. Vasseur, N. Papanikolaou, S. Bench-abane, V. Laude, and A. Martinez, “Simultaneous existence of phononic and photonic band gaps in periodic crystal slabs,” Opt. Express 18(13), 14301–14310 (2010). [CrossRef] [PubMed]

16.

D. Chang, A. H. Safavi-Naeini, M. Hafezi, and O. Painter, “Slowing and stopping light using an optomechanical crystal array,” arXiv:1006.3829 (2010).

17.

A. H. Safavi-Naeini and O. Painter, “Proposal for an Optomechanical Traveling Wave Phonon-Photon Translator,” New J. Phys. 13, 013017 (2011). [CrossRef]

18.

R. H. Stolen and E. P. Ippen, “Raman gain in glass optical waveguides,” Appl. Phys. Lett. 22(276), 276–278 (1973). [CrossRef]

19.

X. Zhang, R. Sooryakumar, and K. Bussmann , “Confinement and transverse standing acoustic resonances in free-standing membranes,” Phys. Rev. B 68, 115430 (2003). [CrossRef]

20.

W. Cheng, N. Gomopoulos, G. Fytas, T. Gorishnyy, J. Walish, E. L. Thomas, A. Hiltner, and E. Bae, “Phonon Dispersion and Nanomechanical Properties of Periodic 1D Multilayer Polymer Films,” Nano Lett. 8(5), 1423–8 (2008). [CrossRef] [PubMed]

21.

T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, “Analysis of Radiation-Pressure Induced Mechanical Oscillation of an Optical Microcavity,” Phys. Rev. Lett. 95(3), 033,901–033,901 (2005). [CrossRef]

22.

Q. Lin, J. Rosenberg, X. Jiang, K. J. Vahala, and O. Painter, “Mechanical Oscillation and Cooling Actuated by the Optical Gradient Force,” Phys. Rev. Lett. 103(10), 103601 (2009). [CrossRef] [PubMed]

23.

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459(7246), 550–555 (2009). [CrossRef] [PubMed]

24.

G. S. Wiederhecker, A. Brenn, H. L. Fragnito, and P. S. J. Russell, “Coherent Control of Ultrahigh-Frequency Acoustic Resonances in Photonic Crystal Fibers,” Phys. Rev. Lett. 100(20), 203903 (2008). [CrossRef] [PubMed]

25.

K. L. Ekinci and M. L. Roukes, “Nanoelectromechanical systems,” Rev. Sci. Instrum. 76(6), 061101–061101 (2005). [CrossRef]

26.

C. T. C. Nguyen, “MEMS technology for timing and frequency control,” IEEE Trans Ultrason Ferroelectr Freq Control 54(2), 251–270 (2007). [CrossRef] [PubMed]

27.

H. Campanella, Acoustic Wave and Electromechanical Resonators: Concept to Key Applications (Artech House Publishers, 2010).

28.

B. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater 4(3), 207–210 (2005). [CrossRef]

29.

M. Borselli, T. J. Johnson, and O. Painter, “Measuring the role of surface chemistry in silicon microphotonics,” Appl. Phys. Lett. 88(13), 131,114–3 (2006). [CrossRef]

30.

COMSOL Multphysics 3.5 (2009).

31.

A. Akhieser, J. Phys. (Moscow) 1, 277 (1939).

32.

L. Landau and G. Rumer, “Absorption of sound in solids,” Phys. Z. Sowjetunion 11, 18 (1937).

33.

C. Zener, “Internal Friction in Solids II. General Theory of Thermoelastic Internal Friction,” Phys. Rev. 53(1), 90 (1938). [CrossRef]

34.

S. D. Lambade, G. G. Sahasrabudhe, and S. Rajagopalan, “Temperature dependence of acoustic attenuation in silicon,” Phys. Rev. B 51(22), 15,861 (1995). [CrossRef]

35.

C. Zener, “Internal Friction in Solids. I. Theory of Internal Friction in Reeds,” Phys. Rev. 52(3), 230 (1937). [CrossRef]

36.

R. Lifshitz and M. L. Roukes, “Thermoelastic damping in micro- and nanomechanical systems,” Phys. Rev. B 61(8), 5600 (2000). [CrossRef]

37.

T. O. Woodruff and H. Ehrenreich, “Absorption of Sound in Insulators,” Phys. Rev. 123(5), 1553 (1961). [CrossRef]

38.

J. Philip and M. A. Breazeale, “Third-order elastic constants and Grüneisen parameters of silicon and germanium between 3 and 300 K,” J. Appl. Phys. 54(2), 752 (1983). [CrossRef]

39.

S. K. Estreicher, M. Sanati, D. West, and F. Ruymgaart, “Thermodynamics of impurities in semiconductors,” Phys. Rev. B 70(12), 125209 (2004). [CrossRef]

40.

A. Duwel, R. Candler, T. Kenny, and M. Varghese, “Engineering MEMS Resonators With Low Thermoelastic Damping,” J. Microelectromech. Syst. 15(6), 1437–1445 (2006). [CrossRef]

OCIS Codes
(220.4880) Optical design and fabrication : Optomechanics
(230.1040) Optical devices : Acousto-optical devices
(230.5298) Optical devices : Photonic crystals

ToC Category:
Photonic Crystals

History
Original Manuscript: December 21, 2010
Revised Manuscript: February 21, 2011
Manuscript Accepted: February 22, 2011
Published: March 11, 2011

Citation
Thiago P. Mayer Alegre, Amir Safavi-Naeini, Martin Winger, and Oskar Painter, "Quasi-two-dimensional optomechanical crystals with a complete phononic bandgap," Opt. Express 19, 5658-5669 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-6-5658


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References

  1. Z. Liu, X. Zhang, Y. Mao, Y. Y. Zhu, Z. Yang, C. T. Chan, and P. Sheng, “Locally Resonant Sonic Materials,” Science 289(5485), 1734–1736 (2000). [CrossRef] [PubMed]
  2. S. Yang, J. H. Page, Z. Liu, M. L. Cowan, C. T. Chan, and P. Sheng, “Ultrasound Tunneling through 3D Phononic Crystals,” Phys. Rev. Lett. 88(10), 104301 (2002). [CrossRef] [PubMed]
  3. W. Cheng, J. Wang, U. Jonas, G. Fytas, and N. Stefanou, “Observation and tuning of hypersonic bandgaps in colloidal crystals,” Nat. Mater. 5(10), 830–836 (2006). [CrossRef] [PubMed]
  4. A. V. Akimov, Y. Tanaka, A. B. Pevtsov, S. F. Kaplan, V. G. Golubev, S. Tamura, D. R. Yakovlev, and M. Bayer, “Hypersonic Modulation of Light in Three-Dimensional Photonic and Phononic Band-Gap Materials,” Phys. Rev. Lett. 101(3), 033902 (2008). [CrossRef] [PubMed]
  5. T. Gorishnyy, C. K. Ullal, M. Maldovan, G. Fytas, and E. L. Thomas, “Hypersonic Phononic Crystals,” Phys. Rev. Lett. 94(11), 115501 (2005). [CrossRef] [PubMed]
  6. S. Mohammadi, A. A. Eftekhar, W. D. Hunt, and A. Adibi, “High-Q micromechanical resonators in a twodimensional phononic crystal slab,” Appl. Phys. Lett. 94(5), 051906 (2009). [CrossRef]
  7. Y. Wen, J. Sun, C. Dais, D. Grtzmacher, T. Wu, J. Shi, and C. Sun, “Three-dimensional phononic nanocrystal composed of ordered quantum dots,” Appl. Phys. Lett. 96(12), 123113 (2010). [CrossRef]
  8. S. Yang, J. H. Page, Z. Liu, M. L. Cowan, C. T. Chan, and P. Sheng, “Focusing of Sound in a 3D Phononic Crystal,” Phys. Rev. Lett. 93(2), 024301 (2004). [CrossRef] [PubMed]
  9. J. Gaofeng and S. Zhifei, “A new seismic isolation system and its feasibility study,” Earthq. Eng. Eng. Vib. 9(1), 75–82 (2010). [CrossRef]
  10. M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462(7269), 78–82 (2009). [CrossRef] [PubMed]
  11. M. Maldovan and E. L. Thomas, “Simultaneous localization of photons and phonons in two-dimensional periodic structures,” Appl. Phys. Lett. 88(25), 251907 (2006). [CrossRef]
  12. M. Eichenfield, J. Chan, A. H. Safavi-Naeini, K. J. Vahala, and O. Painter, “Modeling dispersive coupling and losses of localized optical and mechanical modes in optomechanical crystals,” Opt. Express 17(22), 20078–20098 (2009). [CrossRef] [PubMed]
  13. S. Mohammadi, A. A. Eftekhar, A. Khelif, and A. Adibi, “Simultaneous two-dimensional phononic and photonic band gaps in opto-mechanical crystal slabs,” Opt. Express 18(9), 9164–9172 (2010). [CrossRef] [PubMed]
  14. A. H. Safavi-Naeini and O. Painter, “Design of optomechanical cavities and waveguides on a simultaneous bandgap phononic-photonic crystal slab,” Opt. Express 18(14), 14926–14943 (2010). [CrossRef] [PubMed]
  15. Y. Pennec, B. D. Rouhani, E. H. El Boudouti, C. Li, Y. El Hassouani, J. O. Vasseur, N. Papanikolaou, S. Benchabane, V. Laude, and A. Martinez, “Simultaneous existence of phononic and photonic band gaps in periodic crystal slabs,” Opt. Express 18(13), 14301–14310 (2010). [CrossRef] [PubMed]
  16. D. Chang, A. H. Safavi-Naeini, M. Hafezi, and O. Painter, “Slowing and stopping light using an optomechanical crystal array,” arXiv:1006.3829 (2010).
  17. A. H. Safavi-Naeini and O. Painter, “Proposal for an Optomechanical Traveling Wave Phonon-Photon Translator,” N. J. Phys. 13, 013017 (2011). [CrossRef]
  18. R. H. Stolen and E. P. Ippen, “Raman gain in glass optical waveguides,” Appl. Phys. Lett. 22(276), 276–278 (1973). [CrossRef]
  19. X. Zhang, R. Sooryakumar, and K. Bussmann, “Confinement and transverse standing acoustic resonances in free-standing membranes,” Phys. Rev. B 68, 115430 (2003). [CrossRef]
  20. W. Cheng, N. Gomopoulos, G. Fytas, T. Gorishnyy, J. Walish, E. L. Thomas, A. Hiltner, and E. Bae, “Phonon Dispersion and Nanomechanical Properties of Periodic 1D Multilayer Polymer Films,” Nano Lett. 8(5), 1423–1428 (2008). [CrossRef] [PubMed]
  21. T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, “Analysis of Radiation-Pressure Induced Mechanical Oscillation of an Optical Microcavity,” Phys. Rev. Lett. 95(3), 033,901–033,901 (2005). [CrossRef]
  22. Q. Lin, J. Rosenberg, X. Jiang, K. J. Vahala, and O. Painter, “Mechanical Oscillation and Cooling Actuated by the Optical Gradient Force,” Phys. Rev. Lett. 103(10), 103601 (2009). [CrossRef] [PubMed]
  23. M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photoniccrystal optomechanical cavity,” Nature 459(7246), 550–555 (2009). [CrossRef] [PubMed]
  24. G. S. Wiederhecker, A. Brenn, H. L. Fragnito, and P. S. J. Russell, “Coherent Control of Ultrahigh-Frequency Acoustic Resonances in Photonic Crystal Fibers,” Phys. Rev. Lett. 100(20), 203903 (2008). [CrossRef] [PubMed]
  25. K. L. Ekinci and M. L. Roukes, “Nanoelectromechanical systems,” Rev. Sci. Instrum. 76(6), 061101–061101 (2005). [CrossRef]
  26. C. T. C. Nguyen, “MEMS technology for timing and frequency control,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 54(2), 251–270 (2007). [CrossRef] [PubMed]
  27. H. Campanella, Acoustic Wave and Electromechanical Resonators: Concept to Key Applications (Artech House Publishers, 2010).
  28. B. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. 4(3), 207–210 (2005). [CrossRef]
  29. M. Borselli, T. J. Johnson, and O. Painter, “Measuring the role of surface chemistry in silicon microphotonics,” Appl. Phys. Lett. 88(13), 131,114–3 (2006). [CrossRef]
  30. COMSOL Multphysics 3.5 (2009).
  31. A. Akhieser, J. Phys. (Moscow) 1, 277 (1939).
  32. L. Landau and G. Rumer, “Absorption of sound in solids,” Phys. Z. Sowjetunion 11, 18 (1937).
  33. C. Zener, “Internal Friction in Solids II. General Theory of Thermoelastic Internal Friction,” Phys. Rev. 53(1), 90 (1938). [CrossRef]
  34. S. D. Lambade, G. G. Sahasrabudhe, and S. Rajagopalan, “Temperature dependence of acoustic attenuation in silicon,” Phys. Rev. B 51(22), 15,861 (1995). [CrossRef]
  35. C. Zener, “Internal Friction in Solids. I. Theory of Internal Friction in Reeds,” Phys. Rev. 52(3), 230 (1937). [CrossRef]
  36. R. Lifshitz and M. L. Roukes, “Thermoelastic damping in micro- and nanomechanical systems,” Phys. Rev. B 61(8), 5600 (2000). [CrossRef]
  37. T. O. Woodruff and H. Ehrenreich, “Absorption of Sound in Insulators,” Phys. Rev. 123(5), 1553 (1961). [CrossRef]
  38. J. Philip and M. A. Breazeale, “Third-order elastic constants and Gr¨uneisen parameters of silicon and germanium between 3 and 300 K,” J. Appl. Phys. 54(2), 752 (1983). [CrossRef]
  39. S. K. Estreicher, M. Sanati, D. West, and F. Ruymgaart, “Thermodynamics of impurities in semiconductors,” Phys. Rev. B 70(12), 125209 (2004). [CrossRef]
  40. A. Duwel, R. Candler, T. Kenny, and M. Varghese, “Engineering MEMS Resonators With Low Thermoelastic Damping,” J. Microelectromech. Syst. 15(6), 1437–1445 (2006). [CrossRef]

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